This post-processing hence allows one to apply Theorem REF  even if the collected data does not satisfy O2. One can think of the secondary preparations as noisy versions of the primary preparations. Hence, the price one pays in this construction is that the corresponding noise parameters \(\epsilon ^{\prime }_s = p(M_s|P^{(2)}_s)\)  in O1ni will in general be larger. Note that, even if \(\epsilon ^{\prime }_s\)  is too large compared to \(\epsilon _s\)  to see any violation in Theorem REF , one can get around this issue by adding extra experimental preparations \(P^{(1)}_{\rm extra}\)  to enlarge \(\mathcal {C}\) , as explicitly done in Ref. [1]}. To summarize, there are good general tools to deal with imperfections in the operational equivalences O2.
